Finding an Unique and “Natural” Extension of the Expected Value That Is Finite for All Functions in Non-Shy Subset of the Set of All Measurable Functions
Bharath S. Krishnan · Preprints.org · 2024
Suppose for n∈ ℕ, set A⊆ℝⁿ and function f∶A→ℝⁿ. If set A is Borel; we want to find an unique and “natural” extension of the expected value, w.r.t the Hausdorff measure on bounded functions to unbounded/bounded f, that's finite for all f in a non-shy subset of B*---the set of all Borel measurable functions in ℝᴬ. The issue is current extensions of the expected value are finite for all functions in only a shy subset of B*. The reason this issue wasn't resolved is mathematicians have not thought of the problem, focusing on application rather than generalization. Despite the lack of potential use, we'll attempt to solve the problem by defining a choice function---this shall choose a unique set of equivalent sequences of sets (Fₖ***) , where the set-theoretic limit of Fₖ*** is the graph of f; the measure Hʰ is the ℎ-Hausdorff measure, such for each k∈ℕ, 0 0)∃(N∈ℕ)∀(k∈N)(k≥N⇒1/Hʰ(dom(Fₖ***))∫dom(Fₖ*** )fₖ* dHʰ - E**[f,Fₖ***]<ε ) which should be unique and “natural” extension for all f in a non-shy subset of B*. Note we guessed the choice function using computer programming but we don’t use mathematical proofs due to the lack of expertise in the subject matter. Despite this, the biggest use of this research is the extension of the expected value is finite for "almost all" functions: this is easier to use in application when finding the "average" of functions covering an infinite expanse of space.